A paradox arises when two seemingly airtight arguments lead to contradictory conclusions—conclusions that cannot possibly both be true. It’s similar to adding a set of numbers in a two-dimensional array and getting different answers depending on whether you sum up the rows first or the columns. Since the correct total must be the same either way, the difference shows that an error must have been made in at least one of the two sets of calculations. But it remains to discover at which step (or steps) an erroneous calculation occurred in either or both of the running sums. There are two ways to rebut an argument. We might call them countering and invalidating.

+To counter an argument is to provide another argument that establishes the opposite conclusion.

+To invalidate an argument, we show that there is some step in that argument that simply does not follow from what precedes it (or we show that the argument’s premises—the initial steps—are themselves false).

If an argument starts with true premises, and if every step in the argument does follow, then the argument’s conclusion must be true. However, invalidating an argument—identifying an incorrect step somewhere—does not show that the argument’s conclusion must be false. Rather, the invalidation merely removes that argument itself as a reason to think the conclusion true; the conclusion might still be true for other reasons. Therefore, to firmly rebut an argument whose conclusion is false, we must both invalidate the argument and also present a counterargument for the opposite conclusion.

In the case of a paradox, invalidating is especially important. Whichever of the contradictory conclusions is incorrect, we’ve already got an argument to counter it—that’s what makes the matter a paradox in the first place! Piling on additional counterarguments may (or may not) lead to helpful insights, but the counterarguments themselves cannot suffice to resolve the paradox. What we must also do is invalidate the argument for the false conclusion—that is, we must show how that argument contains one or more steps that do not follow.

Failing to recognize the need for invalidation can lead to frustratingly circular exchanges between proponents of the conflicting positions. One side responds to the other’s argument with a counterargument, thinking it a sufficient rebuttal. The other side responds with a counter- counterargument—perhaps even a repetition of the original argument— thinking it an adequate rebuttal of the rebuttal. This cycle may persist indefinitely. With due attention to the need to invalidate as well as counter, we can interrupt the cycle and achieve a more productive discussion.

Gary Drescher (Good and Real)